Subgroups — #23766356

Order 10112.

480 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
2	#19098470	C₂	1	1	yes	maximal, characteristic
3	#19098448	C₂	1	2	yes	maximal
4	#19098446	C₂	1	2	yes	maximal
5	#19098432	C₂	1	1	yes	maximal, characteristic
6	#19098427	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₇₉⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
10	#19098414	C₂	1	2	yes	maximal
11	#391×C₇₉	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₇₉	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₇₉	C₂²	1	2	yes	—
16	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₇₉⋊C₄	C₂²	1	2	yes	—
19	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
22	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
23	C₂×C₄×Dic₇₉	C₂²	1	2	yes	—
24	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
25	C₇₉⋊#95	C₂²	1	2	yes	—
26	C₇₉⋊#95	C₂²	1	2	yes	—
27	C₇₉⋊#95	C₂²	1	1	yes	characteristic
28	C₇₉⋊#95	C₂²	1	1	yes	characteristic
29	C₇₉⋊#95	C₂²	1	2	yes	—
30	C₇₉⋊#95	C₂²	1	2	yes	—
31	C₂²×Q₈×C₇₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₇₉	C₂²	1	1	yes	characteristic
34	#95×C₇₉	C₂²	1	2	yes	—
35	#95×C₇₉	C₂²	1	2	yes	—
36	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₇₉	C₂³	1	2	yes	—
38	C₂²×Dic₇₉	C₂³	1	2	yes	—
39	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₇₉	—	2	2	no	—
43	C₄⋊Dic₇₉	C₂×C₄	1	8	yes	—
44	C₄⋊Dic₇₉	C₂×C₄	1	4	yes	—
45	Dic₇₉⋊C₄	—	2	4	no	—
46	Dic₇₉⋊C₄	—	2	2	no	—
47	Dic₇₉⋊C₄	—	4	2	no	—
48	Dic₇₉⋊C₄	Q₈	1	4	yes	complemented
49	C₄×Dic₇₉	—	2	2	no	—
50	C₄×Dic₇₉	—	4	2	no	—
51	C₄×Dic₇₉	—	4	2	no	—
52	C₄×Dic₇₉	C₂×C₄	1	8	yes	—
53	C₄×Dic₇₉	C₂×C₄	1	4	yes	—
54	C₂×Q₈×C₇₉	—	2	2	no	—
55	C₂×Q₈×C₇₉	—	2	2	no	—
56	C₂×Q₈×C₇₉	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
59	C₂²×C₄×C₇₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₇₉	—	2	2	no	—
63	C₄²×C₇₉	—	2	2	no	—
64	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
65	Q₈×C₇₉	—	2	8	no	—
66	C₂×C₄×C₇₉	—	2	1	no	—
67	C₂×C₄×C₇₉	—	2	2	no	—
68	C₂×C₄×C₇₉	—	2	2	no	—
69	C₂×C₄×C₇₉	—	2	1	no	—
70	C₂×C₄×C₇₉	—	2	1	no	—
71	C₂×C₄×C₇₉	—	2	2	no	—
72	C₂×C₄×C₇₉	—	2	1	no	—
73	C₂×C₄×C₇₉	—	2	2	no	—
74	C₂×C₄×C₇₉	—	2	2	no	—
75	C₂×C₄×C₇₉	—	2	1	no	—
76	C₂×C₄×C₇₉	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₇₉	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₇₉	#41	1	2	yes	—
80	C₂×C₄×C₇₉	#41	1	2	yes	—
81	C₂×Dic₇₉	—	2	1	no	—
82	C₂×Dic₇₉	—	2	2	no	—
83	C₂×Dic₇₉	—	2	1	no	—
84	C₂×Dic₇₉	—	2	1	no	—
85	C₂×Dic₇₉	—	2	2	no	—
86	C₂×Dic₇₉	—	2	1	no	—
87	C₂×Dic₇₉	—	2	1	no	—
88	C₂×Dic₇₉	—	2	1	no	—
89	C₂×Dic₇₉	—	2	2	no	—
90	C₂×Dic₇₉	—	2	2	no	—
91	C₂×Dic₇₉	—	2	2	no	—
92	C₂×Dic₇₉	—	2	1	no	—
93	C₂×Dic₇₉	C₂²×C₄	1	4	yes	—
94	C₂×Dic₇₉	C₂²×C₄	1	2	yes	—
95	C₂×Dic₇₉	C₂²×C₄	1	2	yes	—
96	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
97	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
98	C₂×Dic₇₉	#41	1	2	yes	—
99	C₂²×C₇₉	C₂³×C₄	1	1	yes	characteristic
100	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
103	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
104	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂²×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
106	C₄×C₇₉	—	2	4	no	—
107	C₄×C₇₉	—	2	2	no	—
108	C₄×C₇₉	—	4	1	no	—
109	C₄×C₇₉	—	4	2	no	—
110	C₄×C₇₉	#117	1	4	yes	—
111	Dic₇₉	—	2	4	no	—
112	Dic₇₉	—	2	2	no	—
113	Dic₇₉	—	2	2	no	—
114	Dic₇₉	—	4	1	no	—
115	Dic₇₉	—	4	2	no	—
116	Dic₇₉	C₄×Q₈	1	4	yes	—
117	C₂×C₇₉	C₂×#117	1	1	yes	characteristic
118	C₂×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
119	C₂×C₇₉	#520	1	1	yes	characteristic
120	C₂×C₇₉	#554	1	1	yes	characteristic
121	C₂×C₇₉	#556	1	1	yes	characteristic
122	C₂×C₇₉	#557	1	1	yes	characteristic
123	C₂×C₇₉	#559	1	1	yes	characteristic
124	#2355	—	79	1	no	maximal, Sylow
125	C₇₉	#2355	1	1	yes	characteristic, complemented, Sylow
126	C₂×C₄×Q₈	—	79	1	no	—
127	#391	—	79	1	no	—
128	#391	D₇₉	1	1	yes	characteristic
129	#387	—	79	2	no	—
130	#387	—	79	2	no	—
131	#384	—	79	1	no	—
132	#382	—	79	2	no	—
133	C₄×C₄⋊C₄	—	79	1	no	—
134	C₄×C₄⋊C₄	—	79	1	no	—
135	C₄×C₄⋊C₄	—	79	1	no	—
136	C₄×C₄⋊C₄	—	79	2	no	—
137	C₂²×Q₈	C₂×D₇₉	1	1	yes	characteristic
138	C₄×Q₈	—	79	8	no	—
139	C₂×C₄⋊C₄	—	79	2	no	—
140	C₂×C₄⋊C₄	—	79	1	no	—
141	C₂×C₄⋊C₄	—	79	2	no	—
142	C₂×C₄⋊C₄	—	79	1	no	—
143	C₂×C₄⋊C₄	—	79	1	no	—
144	C₂×C₄⋊C₄	—	79	2	no	—
145	C₂×C₄⋊C₄	—	79	1	no	—
146	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
147	C₂×C₄²	—	79	2	no	—
148	C₂×C₄²	—	79	2	no	—
149	C₂×C₄²	—	79	2	no	—
150	C₂×C₄²	—	79	1	no	—
151	C₂×C₄²	—	79	1	no	—
152	C₂×C₄²	C₂×D₇₉	1	1	yes	characteristic
153	#95	—	79	2	no	—
154	#95	—	79	2	no	—
155	#95	—	79	1	no	—
156	#95	—	79	1	no	—
157	#95	—	79	2	no	—
158	#95	—	79	2	no	—
159	#95	C₂×D₇₉	1	2	yes	—
160	#95	C₂×D₇₉	1	2	yes	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂×Q₈	C₄×D₇₉	1	4	yes	—
164	C₂²×C₄	—	79	1	no	—
165	C₂²×C₄	—	79	1	no	—
166	C₂²×C₄	—	79	2	no	—
167	C₂²×C₄	—	79	1	no	—
168	C₂²×C₄	—	79	1	no	—
169	C₂²×C₄	—	79	2	no	—
170	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
172	C₂²×C₄	C₂²×D₇₉	1	2	yes	—
173	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	316	2	no	—
177	C₄⋊C₄	—	79	8	no	—
178	C₄⋊C₄	—	79	4	no	—
179	C₄⋊C₄	—	79	4	no	—
180	C₄⋊C₄	—	158	2	no	—
181	C₄⋊C₄	—	158	4	no	—
182	C₄⋊C₄	—	158	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	316	2	no	—
185	C₄²	—	316	2	no	—
186	C₄²	—	79	8	no	—
187	C₄²	—	79	4	no	—
188	C₄²	—	158	2	no	—
189	C₂³	C₂³×D₇₉	1	1	yes	characteristic
190	Q₈	—	2	8	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	79	2	no	—
202	C₂×C₄	—	79	2	no	—
203	C₂×C₄	—	79	2	no	—
204	C₂×C₄	—	79	2	no	—
205	C₂×C₄	—	79	4	no	—
206	C₂×C₄	—	79	2	no	—
207	C₂×C₄	—	158	1	no	—
208	C₂×C₄	—	158	2	no	—
209	C₂×C₄	—	158	1	no	—
210	C₂×C₄	—	158	2	no	—
211	C₂×C₄	—	158	2	no	—
212	C₂×C₄	—	158	1	no	—
213	C₂×C₄	—	158	1	no	—
214	C₂×C₄	—	158	1	no	—
215	C₂×C₄	—	158	2	no	—
216	C₂×C₄	—	158	1	no	—
217	C₂×C₄	—	158	2	no	—
218	C₂×C₄	—	158	1	no	—
219	C₂×C₄	C₂×C₄×D₇₉	1	4	yes	—
220	C₂×C₄	C₂×C₄×D₇₉	1	2	yes	—
221	C₂×C₄	Q₈×D₇₉	1	2	yes	—
222	C₂×C₄	#11929623	1	2	yes	—
223	C₂×C₄	#11929623	1	2	yes	—
224	C₂²	C₂²×C₄×D₇₉	1	1	yes	characteristic
225	C₂²	C₂×Q₈×D₇₉	1	1	yes	characteristic
226	C₂²	#41×D₇₉	1	1	yes	characteristic
227	C₂²	C₂×#11929623	1	1	yes	characteristic
228	C₂²	C₂×#11929623	1	1	yes	characteristic
229	C₂²	#14579207	1	1	yes	characteristic
230	C₂²	#14579209	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	4	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	316	1	no	—
236	C₄	—	316	2	no	—
237	C₄	—	79	4	no	—
238	C₄	—	158	2	no	—
239	C₄	—	158	4	no	—
240	C₄	—	158	2	no	—
241	C₄	#14579117	1	4	yes	—
242	C₂	C₂×#14579117	1	1	yes	characteristic
243	C₂	C₄×Q₈×D₇₉	1	1	yes	characteristic
244	C₂	#19099045	1	1	yes	characteristic
245	C₂	#19099271	1	1	yes	characteristic
246	C₂	#19099284	1	1	yes	characteristic
247	C₂	#19099288	1	1	yes	characteristic
248	C₂	#19099295	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂³×C₄
Fitting	#391×C₇₉	C₂
socle, prime radical, cosocle	C₂³×C₇₉	C₂⁴
hypercenter	#391	D₇₉
hypocenter	C₇₉	#2355
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #391

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
12. Q₈×Dic₇₉	C₄	semidirect
48. Dic₇₉⋊C₄	Q₈	semidirect
125. C₇₉	#2355	semidirect